Experimental illustrations of pattern-forming phenomena:

1. Experimental illustrations of pattern-forming phenomena: Examples from Rayleigh-Benard convection, Taylor-vortex flow, and electro convection • Guenter Ahlers • Department of Physics • University of California • Santa Barbara CA USA z d DT Q x n = kinematic viscosity Prandtl number e = DT/DTc - 1 k = thermal diffusivity s = n / k

3. k = (q, p) T = Tcond + dT sin(p z) exp i(q x + p y ) exp( s t )

4. Neutral curve

5. e = 0 k = (q, p)

6. Fluctuations Patterns Equilibrium Paramagnet Ferromagnet <dT> Temperature Q = dT sin( p z ) exp[ i ( q x + p y ) ]

7. Fluctuations well below the onset of convection Structure factor = square of the modulus of the Fourier transform of the snapshot Shadowgraph image of the pattern. The sample is viewed from the top.In essence, the method shows the temperature field. p p Snapshot in real space R / Rc = 0.94 Movie by Jaechul Oh

8. dST ~ k2 e = -0.57 -0.68 -0.78 dST ~ k-4 k k Experiment: J. Oh and G.A., cond-mat/0209104. Linear Theory: J. Ortiz de Zarate and J. Sengers, Phys. Rev. E 66, 036305 (2002).

9. C(k, t) = < ST (k, t) ST (k, t+ t) > / < ST2 (k, t) > C = C0 exp( -s(k) t ) -0.14 s(k) e = -0.70 J. Oh, J. Ortiz de Zarate, J. Sengers, and G.A., Phys. Rev. E 69, 021106 (2004).

10. Just above onset, straight rolls are stable. Theory: A. Schluter, D. Lortz, and F. Busse, J. Fluid Mech. 23, 129 (1965). This experiment: K.M.S. Bajaj, N. Mukolobwiez, N. Currier, and G.A., Phys. Rev. Lett. 83, 5282 (1999).

11. DT k F. Busse and R.M. Clever, J. Fluid Mech. 91, 319 (1979); and references therein.

13. Taylor vortex flow First experiments and linear stability analysis by G.I. Taylor in Cambridge

15. time Inner cylinder speed The rigid top and bottom pin the phase of the vortices. They also lead to the formation of a sub-critical Ekman vortex. M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 (1986). G. A., D.S. Cannell, M.A. Dominguez-Lerma, and R. Heinrichs, Physica, 23D, 202 (1986). A.M. Rucklidge and A.R. Champneys, Physica A 191, 282 (2004). In the interior, a vortex pair is lost or gained when the system leaves the stable band of states. Theory: W. Eckhaus, Studies in nonlinear stability theory, Springer, NY, 1965. Experiment: M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 (1986). G. A., D.S. Cannell, M.A. Dominguez-Lerma, and R. Heinrichs, Physica, 23D, 202 (1986).

16. ( k - kc ) / kc M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 1986.

17. At the free upper surface the pinning of the phase is weak and a vortex pair can be gained or lost. The Eckhaus Instability is never reached. Experiment: M. Linek and G.A., Phys. Rev. E 58, 3168 (1998). Theory: M.C. Cross, P.G. Daniels, P.C. Hohenberg, and E.D. Siggia, J. Fluid Mech. 127, 155 (1983).

18. Free upper surface Rigid boundaries

21. Theory: H. Riecke and H.G. Paap, Phys. Rev. A 33, 547 (1986). M.C. Cross, Phys. Rev. A 29, 391 (1984). P.M. Eagles, Phys. Rev. A 31, 1955 (1985). Experiment: M.A. Dominguez-Lerma, D.S. Cannell and G.A., Phys. Rev. A 34, 4956 (1986).

24. Back to Rayleigh-Benard ! Shadowgraph image of the pattern. The sample is viewed from the top. In essence, the method shows the temperature field. Wavenumber Selection by Domain wall

25. J.R. Royer, P. O'Neill, N. Becker, and G.A., Phys. Rev. E 70 , 036313 (2004).

27. Experiment: J. Royer, P. O’Neill, N. Becker, and G.A., Phys. Rev. E 70, 036313 (2004). Theory: J. Buell and I. Catton, Phys. Fluids 29, 1 (1986) A.C. Newell, T. Passot, and M. Souli, J. Fluid Mech. 220, 187 (1990).

28. W†= 0 V. Croquette, Contemp. Phys. 30, 153 (1989). Y. Hu, R. Ecke, and G. A., Phys. Rev. E 48, 4399 (1993); Phys. Rev. E 51, 3263 (1995).

29. W†= 0

30. Movie by N. Becker

31. Movie by Nathan Becker W†= 0 Spiral-defect chaos: S.W. Morris, E. Bodenschatz, D.S. Cannell, and G.A., Phys. Rev. Lett. 71, 2026 (1993).

32. W = 2p f d2/ n d DT Q n = kinematic viscosity Prandtl number e = DT/DTc - 1 k = thermal diffusivity s = n / k

33. Wc < W†= 16 G. Kuppers and D. Lortz, J. Fluid Mech. 35, 609 (1969). R.M. Clever and F. Busse, J. Fluid Mech. 94, 609 (1979). Y.-C. Hu, R. Ecke, and G.A., Phys. Rev. Lett. 74 , 5040 (1995); Y. Hu, R. E. Ecke, and G.A., Phys. Rev. E 55, 6928 (1997) Y. Hu, W. Pesch, G.A., and R.E. Ecke, Phys. Rev. E 58, 5821 (1998). Movies by Nathan Becker

34. Electroconvection in a nematic liquid crystal Planar Alignment Director V = V0 cos( wt ) Convection for V0 > Vc e = (V0 / Vc) 2 - 1 Anisotropic !

35. Oblique rolls zig zag Director

37. X.-L. Qiu + G.A., Phys. Rev. Lett. 94, 087802 (2005)

38. Rayleigh-Benard convection Fluctuations and linear growth rates below onset Rotational invariance Neutral curve Straight rolls above onset Stability range above onset, Busse Balloon Taylor-vortec flow Eckhaus instability Narrower band due to reduced phase pinning at a free surface Wavenumber selection by a ramp in epsilon More Rayleigh-Benard Wavenumber selection by a domain wall Wavenumber determined by skewed-varicose instability Onset of spiral-defect chaos Rayleigh-Benard with rotation Kuepers-Lortz or domain chaos Electro-convection in a nematic Loss of rotational invariance Summary: